Interpolating Wavelets and Difference Wavelets *

This paper first shows that the Cohen-Daubechies-Feauveau (C-D-F) biorthogonal wavelet can be derived from the interpolating wavelet through a lifting process. Its high-pass filter measures the interpolation error of the averaged data. Next, we propose a new wavelet method, called the difference wavelet method, for efficient representation for functions on R. Its analysis part is simply averaging and finite differencing. The corresponding synthesis part involves a finite difference equation, which is solved by the cyclic reduction method to achieve fast transform. The associated wavelets constitute biorthogonal Riesz bases in L 2 (R). Their decay and regularity properties are investigated. A comparison study on the efficiency is'sues with C-D-F wavelet method is performed. The comparison includes (i) operation counts. for performing wavelet transform, (ii) the approximate power (i.e. the coefficient in the approximation error estimate), and (iii) the compression ratio (by numerical experiments). The results show that the difference wavelet method is about twice more efficient than the C-D-F wavelet method. This efficiency is due to that the difference wavelets are more regular with just slightly bigger essential support than those of C-D-F wavelets.